Contracting Transport Maps on Riemannian Manifolds

[2607.27711] Contracting Transport Maps on Riemannian Manifolds Skip to main content Search Submit Donate Log in Mathematics > Differential Geometry arXiv:2607.27711 (math) [Submitted on 30 Jul 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: Contracting Transport Maps on Riemannian Manifolds Authors: Shrey Aryan , Bang-Xian Han , Zhuo-Nan Zhu View PDF HTML (experimental) Abstract: We use inverse mean curvature flow to construct bi-Lipschitz maps that preserve normalized volume and decrease distances. These maps send a round sphere onto any smooth closed strictly convex hypersurface in a sphere and a flat disk onto any smooth strictly convex free-boundary disk in a Euclidean ball. In dimension two, this proves a conjecture of E. Milman for every smooth two-sphere with Gaussian curvature at least one and gives an analogous result for nonnegatively curved disks whose boundary has geodesic curvature one. The spherical result proves the two-dimensional case of the spectral comparison conjectured by Colding and Minicozzi. Counterexamples in dimensions $n\geq3$ show that the restriction to dimension two is sharp. Furthermore, an equivariant extension of this construction yields, for every $n\geq2$, a contracting transport map from the uniform probability measure on a round hemisphere to the uniform probability measure on any closed geodesically convex subset of positive volume. This settles the remaining uniform-target case of a question raised by Beck and Jerison. In dimension two, we also find geometric conditions under which the uniform measure on the hemisphere can be transported by a contracting map to a broad class of nonuniform probability measures supported on domains in a hemisphere. Comments: This new version includes some new results from arXiv:2605.24705v2 Subjects: Differential Geometry (math.DG) ; Metric Geometry (math.MG) MSC classes: 53C21, 53E10, 49Q22, 53C45, 58J50 Cite as: arXiv:2607.27711 [math.DG] (or arXiv:2607.27711v2 [math.DG] for this version) https://doi.org/10.48550/arXiv.2607.27711 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Bangxian Han [ view email ] [v1] Thu, 30 Jul 2026 05:45:40 UTC (21 KB) [v2] Thu, 27 Aug 2026 02:45:53 UTC (34 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2026-07 Change to browse by: math math.MG References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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